Oscillation
Not to be confused with Ocellation.
“Oscillator” redirects here. For other uses, see Oscillator
(disambiguation).
Oscillation is the repetitive variation, typically in time,
also in dynamic systems in virtually every area of science: for example the beating human heart, business
cycles in economics, predator-prey population cycles in
ecology, geothermal geysers in geology, vibrating strings
in musical instruments, periodic firing of nerve cells in
the brain, and the periodic swelling of Cepheid variable
stars in astronomy.
1 Simple harmonic oscillator
Main article: Simple harmonic motion
The simplest mechanical oscillating system is a weight
attached to a linear spring subject to only weight and
tension. Such a system may be approximated on an air
table or ice surface. The system is in an equilibrium state
when the spring is static. If the system is displaced from
the equilibrium, there is a net restoring force on the mass,
tending to bring it back to equilibrium. However, in moving the mass back to the equilibrium position, it has acquired momentum which keeps it moving beyond that
position, establishing a new restoring force in the opposite sense. If a constant force such as gravity is added to
the system, the point of equilibrium is shifted. The time
taken for an oscillation to occur is often referred to as the
oscillatory period.
Systems where the restoring force on a body is directly
proportional to its displacement, such as the dynamics of
the spring-mass system, are described mathematically by
the simple harmonic oscillator and the regular periodic
motion is known as simple harmonic motion. In the
spring-mass system, oscillations occur because, at the
static equilibrium displacement, the mass has kinetic energy which is converted into potential energy stored in
the spring at the extremes of its path. The spring-mass
system illustrates some common features of oscillation,
namely the existence of an equilibrium and the presence
of a restoring force which grows stronger the further the
system deviates from equilibrium.
An undamped spring–mass system is an oscillatory system
of some measure about a central value (often a point
of equilibrium) or between two or more different states. 2 Damped and driven oscillations
The term 'vibration' is precisely used to describe mechanical oscillation but used as a synonym of 'oscillation' Main article: Harmonic oscillator
too. Familiar examples include a swinging pendulum and
alternating current power.
All real-world oscillator systems are thermodynamically
Oscillations occur not only in mechanical systems but irreversible. This means there are dissipative processes
1
2
4 CONTINUOUS SYSTEMS – WAVES
such as friction or electrical resistance which continually
convert some of the energy stored in the oscillator into
heat in the environment. This is called damping. Thus,
oscillations tend to decay with time unless there is some
net source of energy into the system. The simplest description of this decay process can be illustrated by oscillation decay of the harmonic oscillator.
In addition, an oscillating system may be subject to some
external force, as when an AC circuit is connected to an
outside power source. In this case the oscillation is said
to be driven.
Some systems can be excited by energy transfer from the
environment. This transfer typically occurs where systems are embedded in some fluid flow. For example, the
phenomenon of flutter in aerodynamics occurs when an
arbitrarily small displacement of an aircraft wing (from
its equilibrium) results in an increase in the angle of attack
of the wing on the air flow and a consequential increase in
lift coefficient, leading to a still greater displacement. At
sufficiently large displacements, the stiffness of the wing
dominates to provide the restoring force that enables an
oscillation.
3
Coupled oscillations
Experimental Setup of Huygens synchronization of two clocks
to synchronise. This phenomenon was first observed by
Christiaan Huygens in 1665.[1] The apparent motions of
the compound oscillations typically appears very complicated but a more economic, computationally simpler and
conceptually deeper description is given by resolving the
motion into normal modes.
Two pendulums with the same period fixed on a string act as pair
of coupled oscillators. The oscillation alternates between the two.
The harmonic oscillator and the systems it models have
a single degree of freedom. More complicated systems
have more degrees of freedom, for example two masses
and three springs (each mass being attached to fixed
points and to each other). In such cases, the behavior of
each variable influences that of the others. This leads to
a coupling of the oscillations of the individual degrees of
freedom. For example, two pendulum clocks (of identical frequency) mounted on a common wall will tend
More special cases are the coupled oscillators where energy alternates between two forms of oscillation. Wellknown is the Wilberforce pendulum, where the oscillation alternates between an elongation of a vertical spring
and the rotation of an object at the end of that spring.
4 Continuous systems – waves
Main article: Waves
As the number of degrees of freedom becomes arbitrarily
large, a system approaches continuity; examples include
a string or the surface of a body of water. Such systems
3
have (in the classical limit) an infinite number of normal
modes and their oscillations occur in the form of waves
that can characteristically propagate.
5
Mathematics
Main article: Mathematics of oscillation
The mathematics of oscillation deals with the quantifi-
xn
lim sup
lim inf
n
Oscillation of a sequence (shown in blue) is the difference between the limit superior and limit inferior of the sequence.
cation of the amount that a sequence or function tends
to move between extremes. There are several related notions: oscillation of a sequence of real numbers, oscillation of a real valued function at a point, and oscillation of
a function on an interval (or open set).
6
Examples
7
See also
8
References
[1] Strogatz, Steven. Sync: The Emerging Science of Spontaneous Order. Hyperion, 2003, pp 106-109
9
External links
• Vibrations – a chapter from an online textbook
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TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES
Text and image sources, contributors, and licenses
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